Hyperbolic groups with planar boundaries

TL;DR

This paper proves the quasi-isometric rigidity of convex-cocompact Kleinian groups and characterizes hyperbolic groups with planar boundaries under certain conditions, linking geometric actions to boundary properties.

Contribution

It establishes conditions under which hyperbolic groups with planar boundaries are virtually Kleinian groups, extending rigidity results to new boundary cases.

Findings

Convex-cocompact Kleinian groups are quasi-isometrically rigid.

Hyperbolic groups with planar boundaries and specific boundary dimensions are virtually Kleinian.

Certain geometric actions imply boundary conformal properties.

Abstract

We prove that the class of convex-cocompact Kleinian groups is quasi-isometrically rigid. We also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex-cocompact Kleinian group provided that its boundary has Ahlfors regular conformal dimension strictly less than $2$ or if it acts geometrically on a CAT(0) cube complex.